Physics-informed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multi-scale solutions. Finite basis physics-informed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multi-scale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.

翻译：物理信息神经网络（PINNs）是求解微分方程问题的强大方法，但其在处理具有高频和/或多尺度解的问题时常面临困难。有限基物理信息神经网络（FBPINNs）通过将PINNs与重叠域分解方法相结合，提升了此类场景下的性能。本文受经典多级Schwarz域分解方法（DDMs）启发，通过在FBPINNs的解假设中引入多级域分解结构，对其进行了扩展。类比于经典DDMs的典型测试，我们通过执行强扩展与弱扩展实验，评估了PINNs、FBPINNs及多级FBPINNs的精度如何随计算开销与解复杂度变化而扩展。数值结果表明，所提出的多级FBPINNs在一系列具有高频及多尺度解的问题中，均持续且显著优于PINNs。此外，正如经典DDMs所预期，我们证明了多级FBPINNs通过促进子域间的全局通信，在使用大量子域时提升了FBPINNs的精度。